MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  19.21h Structured version   Visualization version   GIF version

Theorem 19.21h 2296
Description: Theorem 19.21 of [Margaris] p. 90. The hypothesis can be thought of as "𝑥 is not free in 𝜑". See also 19.21 2242 and 19.21v 2031. (Contributed by NM, 1-Aug-2017.) (Proof shortened by Wolf Lammen, 1-Jan-2018.)
Hypothesis
Ref Expression
19.21h.1 (𝜑 → ∀𝑥𝜑)
Assertion
Ref Expression
19.21h (∀𝑥(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝜓))

Proof of Theorem 19.21h
StepHypRef Expression
1 19.21h.1 . . 3 (𝜑 → ∀𝑥𝜑)
21nf5i 2190 . 2 𝑥𝜑
3219.21 2242 1 (∀𝑥(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wal 1635
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-10 2185  ax-12 2214
This theorem depends on definitions:  df-bi 198  df-ex 1860  df-nf 1864
This theorem is referenced by:  hbim1  2301
  Copyright terms: Public domain W3C validator